If tan `theta + cot theta = 2` then the value of `tan^2 theta + cot^2 theta` is

Options

A. `2`
B. `1`
C. `sqrt(2)`
D. `0`

Answer

To solve the problem, we need to find the value of \( \tan^2 \theta + \cot^2 \theta \) given that \( \tan \theta + \cot \theta = 2 \).

### Step-by-Step Solution:

1. **Start with the given equation**:
\[ \tan \theta + \cot \theta = 2 \]

2. **Square both sides**:
\[ (\tan \theta + \cot \theta)^2 = 2^2 \]
This expands to:
\[ \tan^2 \theta + 2 \tan \theta \cot \theta + \cot^2 \theta = 4 \]

3. **Recall the identity**:
Since \( \tan \theta \cot \theta = 1 \), we can replace \( 2 \tan \theta \cot \theta \) with \( 2 \):
\[ \tan^2 \theta + 2 + \cot^2 \theta = 4 \]

4. **Rearrange the equation**:
Subtract 2 from both sides:
\[ \tan^2 \theta + \cot^2 \theta = 4 - 2 \]
Simplifying gives:
\[ \tan^2 \theta + \cot^2 \theta = 2 \]

5. **Final result**:
Therefore, the value of \( \tan^2 \theta + \cot^2 \theta \) is:
\[ \boxed{2} \]